Advanced analysis for periodic signals and coefficients. Tune waveform shape, sampling depth, and reconstruction precision. Export results, inspect convergence, and visualize spectrum behavior clearly.
| Parameter | Example Value | Why It Helps |
|---|---|---|
| Waveform Type | Square Wave | Shows odd-harmonic dominance clearly. |
| Amplitude | 1 | Keeps interpretation simple. |
| Period T | 6.2831853072 | Makes the fundamental angular frequency equal to 1. |
| Duty Cycle | 50% | Creates the classic symmetric waveform. |
| Maximum Harmonic N | 25 | Shows reconstruction improvement across many terms. |
| Integration Samples | 2400 | Improves numerical coefficient accuracy. |
| Plot Points | 700 | Produces a smooth comparison graph. |
This implementation evaluates the signal over one period, approximates the integral numerically, then reconstructs the waveform using the selected harmonic limit.
Choose a waveform or select the custom expression option.
Enter amplitude, period, shift, offset, and duty cycle.
Set the maximum harmonic count for the truncated series.
Increase integration samples for stronger numerical stability.
Set plot points to control graph smoothness.
Submit the form to display results above the inputs.
Review cn values, phases, RMS, energy capture, and error metrics.
Export the visible result table with the CSV or PDF buttons.
It estimates the complex Fourier coefficients cₙ for a periodic signal, then rebuilds the waveform using a finite harmonic range. It also reports magnitude, phase, RMS values, numerical error, and energy capture.
Complex Fourier series uses exponential terms e^(inω₀t). Positive and negative indices represent rotating phasors in opposite directions. Together they encode amplitude and phase efficiently, especially for shifted or asymmetric signals.
A larger harmonic limit includes more spectral detail. Smooth waveforms converge quickly, while discontinuous waveforms need more terms. Near jumps, Gibbs overshoot can still appear even when many harmonics are included.
The coefficients are computed numerically. Higher sample counts reduce integration error and improve coefficient accuracy, especially when the signal has sharp transitions, narrow pulses, or high-frequency custom components.
c₀ is the average value of the signal over one full period. For real-valued signals, it usually appears as the DC component and equals the mean level of the waveform.
Yes. Choose Custom Expression and use supported variables such as t, A, T, w, shift, offset, duty, and pi. Standard functions like sin, cos, sqrt, exp, and log are also supported.
For real signals, the reconstructed output should be real. Small imaginary leftovers occur because the calculator truncates the series and estimates coefficients numerically rather than symbolically.
Use CSV when you want spreadsheet analysis or coefficient filtering. Use PDF when you need a clean report for coursework, documentation, sharing, or archiving your harmonic study output.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.